Abstract. The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for testing the existence of a manifold that fits a probability distribution supported in a separable Hilbert space, only using i.i.d samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution P supported on the unit ball of a separable Hilbert space H. Let G(d, V, τ) be the set of submanifolds of the unit ball of H whose volume is at most V and reach (which is the supremum of all r such that any point at a distance less than r has a unique nearest point on the manifold) is at least τ. Let L(M, P) denote mean-squared distance of a random point from the probability distribution P to M. We obtain an algorithm that tests the manifold hypothesis in the following sense.The algorithm takes i.i.d random samples from P as input, and determines which of the following two is true (at least one must be):(The answer is correct with probability at least 1 − δ.

Let K be a polytope in R n defined by m linear inequalities. We give a new Markov Chain algorithm to draw a nearly uniform sample from K. The underlying Markov Chain is the first to have a mixing time that is strongly polynomial when started from a "central" point x 0 . If s is the supremum over all chords pq passing through x 0 of |p−x0| |q−x0| and is an upper bound on the desired total variation distance from the uniform, it is sufficient to take O mn n log(sm) + log 1 steps of the random walk. We use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately. More precisely, suppose Q = {z Bz ≤ 1} contains a point z such that c T z ≥ d and r := sup z∈Q Bz + 1, where B is an m × n matrix. Then, after τ = O mn n ln mr + ln 1 δ steps, the random walk is at a point x τ for which c T x τ ≥ d(1 − ) with probability greater than 1 − δ. The fact that this algorithm has a run-time that is provably polynomial is notable since the analogous deterministic affine algorithm analyzed by Dikin has no known polynomial guarantees.

Kostka numbers and Littlewood-Richardson coefficients appear in combinatorics and representation theory. Interest in their computation stems from the fact that they are present in quantum mechanical computations since Wigner [15]. In recent times, there have been a number of algorithms proposed to perform this task [1-3, 11, 12]. The issue of their computational complexity has received attention in the past, and was raised recently by E. Rassart in [11]. We prove that the problem of computing either quantity is #P-complete. Thus, unless P = N P, which is widely disbelieved, there do not exist efficient algorithms that compute these numbers.

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Let K be a polytope in R n defined by m linear inequalities. We give a new Markov Chain algorithm to draw a nearly uniform sample from K. The underlying Markov Chain is the first to have a mixing time that is strongly polynomial when started from a "central" point x 0 . If s is the supremum over all chords pq passing through x 0 of |p−x0| |q−x0| and is an upper bound on the desired total variation distance from the uniform, it is sufficient to take O mn n log(sm) + log 1 steps of the random walk. We use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately. More precisely, suppose Q = {z Bz ≤ 1} contains a point z such that c T z ≥ d and r := sup z∈Q Bz + 1, where B is an m × n matrix. Then, after τ = O mn n ln mr + ln 1 δ steps, the random walk is at a point x τ for which c T x τ ≥ d(1 − ) with probability greater than 1 − δ. The fact that this algorithm has a run-time that is provably polynomial is notable since the analogous deterministic affine algorithm analyzed by Dikin has no known polynomial guarantees.

We point out that the positivity of a Littlewood-Richardson coefficient c γ α,β for sl n can be decided in strongly polynomial time. This means that the number of arithmetic operations is polynomial in n and independent of the bit lengths of the specifications of the partitions α, β, and γ , and each operation involves numbers whose bitlength is polynomial in n and the bit lengths α, β, and γ .Secondly, we observe that nonvanishing of a generalized Littlewood-Richardson coefficient of any type can be decided in strongly polynomial time assuming an analogue of the saturation conjecture for these types, and that for weights α, β, γ , the positivity of c 2γ 2α,2β can (unconditionally) be decided in strongly polynomial time.

We present a Markov Chain, "Dikin walk", for sampling from a convex body equipped with a self-concordant barrier. This Markov Chain corresponds to a natural random walk with respect to a Riemannian metric defined using the Hessian of the barrier function.For every convex set of dimension n, there exists a self-concordant barrier whose self-concordance parameter is O(n). Consequently, a rapidly mixing Markov Chain of the kind we describe can be defined (but not always be efficiently implemented) on any convex set. We use these results to design an algorithm consisting of a single random walk for optimizing a linear function on a convex set. Using results of Barthe [2] and Bobkov and Houdré [5], on the isoperimetry of products of weighted Riemannian manifolds, we obtain sharper upper bounds on the mixing time of a Dikin walk on products of convex sets than the bounds obtained from a direct application of the Localization Lemma. The results in this paper generalize previous results of [12] from polytopes to spectrahedra and beyond, and improve upon those results in a special case when the convex set is a direct product of lower dimensional convex sets. This Markov Chain like the chain described in [12] is affine-invariant.MSC classification: 65C40, 90C30